# What is the period of f(t)=sin( t / 16 )+ cos( (t)/18 ) ?

Nov 7, 2017

$288 \pi .$

#### Explanation:

Let, $f \left(t\right) = g \left(t\right) + h \left(t\right) , g \left(t\right) = \sin \left(\frac{t}{16}\right) , h \left(t\right) = \cos \left(\frac{t}{18}\right) .$

We know that $2 \pi$ is the Principal Period of both sin, &, cos

functions (funs.).

$\therefore \sin x = \sin \left(x + 2 \pi\right) , \forall x \in \mathbb{R} .$

Replacing $x$ by $\left(\frac{1}{16} t\right) ,$ we have,

$\sin \left(\frac{1}{16} x\right) = \sin \left(\frac{1}{16} x + 2 \pi\right) = \sin \left(\frac{1}{16} \left(t + 32 \pi\right)\right) .$

$\therefore {p}_{1} = 32 \pi$ is a period of the fun. $g$.

Similarly, ${p}_{2} = 36 \pi$ is a period of the fun. $h$.

Here, it would be very important to note that, ${p}_{1} + {p}_{2}$ is not

the period of the fun. $f = g + h .$

In fact, if $p$ will be the period of $f$, if and only if,

$\exists l , m \in \mathbb{N} , \text{ such that, } l {p}_{1} = m {p}_{2} = p \ldots \ldots \ldots \left(\ast\right)$

So, we have to find

$l , m \in \mathbb{N} , \text{ such that, } l \left(32 \pi\right) = m \left(36 \pi\right) , i . e . ,$

$8 l = 9 m .$

Taking, $l = 9 , m = 8 ,$ we have, from $\left(\ast\right) ,$

$9 \left(32 \pi\right) = 8 \left(36 \pi\right) = 288 \pi = p ,$ as the period of the fun. $f$.

Enjoy Maths.!